Calculus Examples

$f (x) = 4 x^{3} - 9 x^{4}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 4 x^{3} - 9 x^{4}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 x^{3} - 9 x^{4} = 0$ .

$4 x^{3} - 9 x^{4} = 0$

Step 1.2.2

Factor $x^{3}$ out of $4 x^{3} - 9 x^{4}$ .

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Step 1.2.2.1

Factor $x^{3}$ out of $4 x^{3}$ .

$x^{3} \cdot 4 - 9 x^{4} = 0$

Step 1.2.2.2

Factor $x^{3}$ out of $- 9 x^{4}$ .

$x^{3} \cdot 4 + x^{3} (- 9 x) = 0$

Step 1.2.2.3

Factor $x^{3}$ out of $x^{3} \cdot 4 + x^{3} (- 9 x)$ .

$x^{3} (4 - 9 x) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{3} = 0$

$4 - 9 x = 0$

Step 1.2.4

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 1.2.4.2

Solve $x^{3} = 0$ for $x$ .

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Step 1.2.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 1.2.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 1.2.4.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.4.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 1.2.5

Set $4 - 9 x$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $4 - 9 x$ equal to $0$ .

$4 - 9 x = 0$

Step 1.2.5.2

Solve $4 - 9 x = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $4$ from both sides of the equation.

$- 9 x = - 4$

Step 1.2.5.2.2

Divide each term in $- 9 x = - 4$ by $- 9$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $- 9 x = - 4$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 4}{- 9}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 4}{- 9}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 9}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{9}$

Step 1.2.6

The final solution is all the values that make $x^{3} (4 - 9 x) = 0$ true.

$x = 0, \frac{4}{9}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{4}{9}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 4 {(0)}^{3} - 9 {(0)}^{4}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot 0^{3} - 9 {(0)}^{4}$

Step 2.2.2

Remove parentheses.

$y = 4 \cdot 0^{3} - 9 \cdot 0^{4}$

Step 2.2.3

Remove parentheses.

$y = 4 {(0)}^{3} - 9 {(0)}^{4}$

Step 2.2.4

Simplify $4 {(0)}^{3} - 9 {(0)}^{4}$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 4 \cdot 0 - 9 {(0)}^{4}$

Step 2.2.4.1.2

Multiply $4$ by $0$ .

$y = 0 - 9 {(0)}^{4}$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 - 9 \cdot 0$

Step 2.2.4.1.4

Multiply $- 9$ by $0$ .

$y = 0 + 0$

Step 2.2.4.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{4}{9}, 0)$

y-intercept(s): $(0, 0)$

Step 4